Since the invention of optical lens, the study of optics has evolved around the refractive index, the fundamental property of all optical materials. The refractive index described by Snell's law affects propagation speed, wavelength, diffraction, energy density, and absorption and emission of light in materials. So far, experimentally realized wideband refractive indices remain below 40, even with intricately designed artificial media.
Herein, we demonstrate a measured index of 1800 or more resulting from a mesoscopic crystal with a dielectric constant of 3 million or more. This gigantic enhancement effect originates from the space-filling curve concept from mathematics. The concept is very tenacious with respect to wideband refractive index. A wideband mega-dielectric according to the present disclosure promises not only enhanced resolution in imaging and lithography and increased fundamental absorption limits in solar energy devices, but also compact, power-efficient components for optical communication and increased performance in many other applications.
There exists a fundamental upper bound on the refractive index of any natural or artificial medium with atomic scale unit cells. For non-magnetic materials, the refractive index (n) is solely determined by the dielectric constant (εr), which in turn is determined by the atomic (or molecular) polarizability and its spatial arrangement.
The volume-averaged polarizability of an ensemble of ideal two-level systems is summarized by the following Equation 1.
                                          ND            2                    ⁡                      (                                          w                t                            -              w                        )                                                ɛ            0                    ⁢                      ℏ            ⁡                          [                                                                    (                                                                  w                        t                                            -                      w                                        )                                    2                                +                                  γ                  2                                            ]                                                          [                  Equation          ⁢                                          ⁢          1                ]            
N: density of the two-level system
D: relevant transition dipole moment
ε0: vacuum permittivity
ℏ: reduced Planck constant
wt: transition frequency between two levels
w: frequency
γ: effective damping factor
For low frequencies (w<<wt), the volume-averaged polarizability of an ensemble of ideal two-level systems is
            ND      2                      ɛ        0            ⁢      ℏ      ⁢                          ⁢              w        t              .For typical N and D of solids,
      ND    2              ɛ      0        ⁢    ℏ    ⁢                  ⁢          w      t      is on the order of unity, which is why the refractive indices of materials remain also on the order of unity. If one can increase this factor by six orders of magnitude, the dielectric constant would increase by the same amount and the refractive index, by three orders. Existing approaches to increase the refractive index are divided into resonant and non-resonant routes. The resonant schemes aim to minimize the factor in the denominator, wt−w, by working near a resonance (wt≈w), using an atomic transition level or an electromagnetic resonance of artificially designed sub-wavelength structures (meta-atoms). In actual systems, the dielectric constant does not diverge on resonance due to various resonance broadening mechanisms that make γ a non-zero value.
As one minimizes these broadening factors, the resulting index becomes larger at the design frequency; but at the same time, it becomes more frequency-dispersive and the index deviates severely even for slightly different frequencies. This makes propagation of a temporal pulse impossible without distortion. This narrowband nature and enhancement-bandwidth trade-off is an intrinsic property of resonance-based designs and presents a fundamental hurdle for practical implementations of those schemes. On the other hand, there was a proposal to increase the index based on quasi-static boundary conditions, which are free from this trade-off relationship and can provide nearly frequency-independent enhancement over a broad bandwidth. In the proposed classical model, the enhancement can be shown to increase to an arbitrarily large value if the spatial gap between metallic inclusions is reduced. However, the experimentally measured values remained below 40 as several practical and theoretical constraints impose upper bounds on the enhancement. These constraints include such as lateral fabrication resolution, dielectric breakdown, and more fundamentally, the breakdown of classical material models at sub-nanometer size gaps. Hence, a vitally different approach is required to enhance the refractive index much beyond the current record.